Chaos theory

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In mathematics, chaos theory describes the behavior of certain dynamical systems – that is, systems whose states evolve with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.

Chaotic dynamics

For a dynamical system to be classified as chaotic, it must have the following properties:[30]

Assign z to z² minus the conjugate of z, plus the original value of the pixel for each pixel, then count how many cycles it took when the absolute value of z exceeds two; inversion (borders are inner set), so that you can see that it threatens to fail that third condition, even if it meets condition two.

Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour. However, it has been shown that the first two conditions in fact imply this one.[31]

Sensitivity to initial conditions is popularly known as the "butterfly effect," so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C. entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some randomness.[32]

All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point.[32][33] Thus, given a time series to test for determinism, one can:

pick a test state;

search the time series for a similar or 'nearby' state; and

compare their respective time evolutions.

Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby state. A deterministic system will have an error that either remains small (stable, regular solution) or increases exponentially with time (chaos). A stochastic system will have a randomly distributed error.[34]

Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state (i.e., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on phase space embedding methods.[35] Typically one chooses an embedding dimension, and investigates the propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you can increase the dimension to obtain a deterministic looking error, then you are done. Though it may sound simple it is not really. One complication is that as the dimension increases the search for a nearby state requires a lot more computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) to find a suitably close candidate. If the embedding dimension (number of measures per state) is chosen too small (less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the dimension too large – the method will work.

When a non-linear deterministic system is attended by external fluctuations, its trajectories present serious and permanent distortions. Furthermore, the noise is amplified due to the inherent non-linearity and reveals totally new dynamical properties. Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate the deterministic part risk failure. Things become worse when the deterministic component is a non-linear feedback system.[36] In presence of interactions between nonlinear deterministic components and noise the resulting nonlinear series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture.[37]

strive to predict 10 tickets for 10 numbers with all winning numbers for Pick6!

strive to predict one key digit for 10 tickets for pick3 6 boxed way win!

In my opinion, I see parallels between chaos theory and lottery. There are those who opine the placement of the ball numbers in the tube(s) above the hopper are to some degree deterministic as to the final outcome. Limited orbits determined by the hopper, etc.

As to, Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?

I do not believe so. Though I do believe in the Butterfly effect as demonstrated in the movie "Sound of Thunder" where the theory is, " For the lack of a nail the kingdom was lost." Movie is well worth renting.

Are you suggesting that lotteries which appear to lack order, purpose and predictability are like certain dynamical systems whose states evolve with time and their behavior can be described mathematically using the chaos theory?

* you don't need more tickets, just the right ticket * * your best chance at winning a lottery jackpot is to buy a ticket * "I will magically reveal the winning numbers after the drawing"

I wish lottery systems could be modeled by chaos theory! than all we have to do is identify initial conditions and create computer simulation and start approximating identifiable singularities or tendencies to form singularities (by let say catastrophe theory), big question would be to "reverse engineer" initial condition of course:) now let say if you even succeeded in doing so, a slight change in the initial condition such a variation of 1/10000 degree F in temperature would immediately invalidate your model:) and thats the essence of chaos theory: ultra sensitivity to the initial conditions

I vote for non-applicability of this model/theory to solve "win the lottery" problem/issue at hand:)

ab actu ad posse valet illatio - from the past one can infer the future

Over time, the lotteries have been more and more inclined to use multiple ball sets and machines, mixing them up, rather than using one and keeping others just to use as spares. Some games have stopped using mechanical systems altogether.

The other thing is that for lotto games, the matricies grow far more often than they shrink. Even a "good" system is harder to develop and costs more to play, under that scenario.

In neo-conned Amerika, bank robs you.Alcohol, Tobacco, and Firearms should be the name of a convenience store, not a govnoment agency.

In order to catch up with random sparks, often random vs random, while anyone's method used to minimize the space in which the allowed random is used to catch the winning random numbers. Lottery evolves through time, so a particular system will go cold while others get hot, vice versa, similar to chaos theory with its evolution. Since both are random, for lotto if its used with machine balls instead of RNG, then that tells us its extremely flexible while our methods in picking numbers are stiff. In order to have a chance, the methods we come up with must be flexible, fluid like while allowing random factor within the walls to catch the winning set, if not the first prize, at least one from the rest.

Chaos theory only applies to the environment which is not closed,open to all future non detected possibilities.and lottery is in a closed environment where each ball has equal chance to come out over a long period of time.

chaos theory says that when the position is fixed,the trace can not be accurately predicted.But this also does apply to the lottery machine as every moment one position can have many balls coming through,so the position in a closed environment can not be fixed.

There are some strategies that could catch a certain group of numbers that cover all 6 numbers,and improves the chance to hit the jackpot though.The least waiting time strategy is the best.

Hans

strive to predict 10 tickets for 10 numbers with all winning numbers for Pick6!

strive to predict one key digit for 10 tickets for pick3 6 boxed way win!

Being educated in fields other than pure mathematics I think there are aspects of Choas Theory that can be applied to the lottory. What those aspects are and how they might be employed against the lottory is still undecided.

I agree most heartedly with DiamondPalace when he states that the lottory is flexible while our systems (or rather the ones we seem to be talking about the most) are not. Choas Theory is not flexible. Initial conditions determine the results and one little iota of change in one variable will adversely affect the results. Well, is that not what the lottory does? As Time*Treat states, they use several different machine with several different ball sets, all mixed up not to mention the fact these machines are moved out of the way between draws, loaded haphazardly, and then moved onto the set a few hours before the draw.

While the application of Chaos Theory against the lottory is in its infancy, my mind tells me there is something there worth pursuing. I would like to see some members devote more threads to this topic. I cerainly intend on doing so.

Any series of random numbers has a pattern to it. Intensive analysis will discover this pattern, but the discovery will be of no use for future predictions.

Consider the possibility that lottery numbers are not random. They must be very close to random, or smart people would discover the pattern.

Consider the possibility that lottery numbers are not random. They must be very close to random, or smart people would discover the pattern.

I believe smart people thinking random lottery numbers have no pattern would not spend any time trying to find a pattern which leaves any such research to the not so smart people. All it would take is for a player to start winning lotteries in such a way that defies the odds for some smart people to become interested in what he's doing.

* you don't need more tickets, just the right ticket * * your best chance at winning a lottery jackpot is to buy a ticket * "I will magically reveal the winning numbers after the drawing"

Dictionary definitions emphasize the notion of "apparent absence of cause, planning, or design," "lack of method or system," or "accidental, haphazard."

The mathematical conceptions of randomness involve deviations from distributions which are infinite in size. No empirical process can be tested against this idealized notion of randomness because we can't collect an infinite number of data points. We can't even judge something as being non-random from a single, weird pattern because random processes will sometimes produce short sequences which appear to be non-random.

Chaos theory has shown that deterministic systems can produce results which are chaotic and appear to be random. But they are not technically random because the events can be modelled by a (non-linear) formula. The classic example of such a system is the pseudo-random number generators used by computers.

Other systems are stable, linear, or non-chaotic under some conditions, but under other conditions do dissolve into randomness or unpredictability (on some level). But even then, on a different level or scale, patterns can still be found. A dripping faucet is an example of such a system. With some water flows, there is a steady a predictable drip, but at other (lower) levels the drips appear to be randomly irregular (but at a higher level of analysis, patterns can be detected, suggesting only chaos), and at still other water levels, no pattern can be discerned at the levels of analysis used for previous drip rates.

So the notion and practice of randomness/chaos is not as simple or cut and dried as people think.

A mind once stretched by a new idea never returns to its original dimensions!

Consider the possibility that lottery numbers are not random. They must be very close to random, or smart people would discover the pattern.

I believe smart people thinking random lottery numbers have no pattern would not spend any time trying to find a pattern which leaves any such research to the not so smart people. All it would take is for a player to start winning lotteries in such a way that defies the odds for some smart people to become interested in what he's doing.

"All it would take is for a player to start winning lotteries in such away that defies the odds for some smart people to become interested inwhat he's doing."

And plenty of not-so-smart and not-so-nice people, too.

In neo-conned Amerika, bank robs you.Alcohol, Tobacco, and Firearms should be the name of a convenience store, not a govnoment agency.